Causality and Quantum Theory

CAUSALITY AND QUANTUM THEORY BLAKE K. WINTER Abstract. We begin with a brief summary of issues encountered involving causality in quantum theory, placing careful emphasis on the assumptions involved in results such as the EPR paradox and Bell’s inequality. We critique some solutions to the resulting paradox, including Rovelli’s relational quantum mechanics and the many-worlds interpretation. We then discuss how a spacetime manifold could come about on the classical level out of a quantum system, by constructing a space with a topology out of the algebra of observables, and show that even with an hypothesis of superluminal causation enforcing consistent measurements of entangled states, a causal cone structure arises on the classical level. Finally, we discuss the possibility that causality as understood in classical relativistic physics may be an emergent symmetry which does not hold on the quantum level. 1. Quantum theory Recall that a quantum system may be described by a Hilbert space H (for technical reasons, this must often be a rigged Hilbert space, [1, 8, 11] together with some subalgebra O of the self-adjoint operators on H. Operators in O correspond to observables. For example, the spin of a spin-1/2 particle is described by a two- dimensional Hilbert space with an orthonormal basis of unit vectors given by (using Dirac’s bra-ket notation) |ui and |di, the up and down states respectively. A state of the system is a unit vector in H. According to the usual rules of quantum mechanics. When an observable A ∈ O is measured, the possible outcomes are given by the spectrum of A. If ai is an eigenvalue for A, and the system is in the state |Ψi then the probability of obtaining ai as the result of a measurement t 2 of A is given by | hΨ| Pai Pai |Ψi | , where Pai is the projection operator onto the eigenspace of A corresponding to the eigenvalue ai. The quantity hΦ| Ψi is called the probability amplitude of finding the system in the state |Φi when we measure it while it is in the state |Ψi. arXiv:1705.07201v1 [quant-ph] 19 May 2017 According to the usual rules of quantum mechanics, a system normally evolves in a unitary fashion. That is, the state |Ψ(t)i is given by U(t) |Ψ(0)i for some unitary function U(t). However, when a system is measured, the state changes in a non-unitary fashion: if a measurement of the observable A while the system is in the state |Ψiyields the result ai, then the system changes to the state Pai |Ψi. This is the usual axiom of measurement in quantum theory. The difficulty in interpreting quantum mechanics is that the resulting probabilities are not additive, as we might expect in classical probability theory; rather, if a state |Ψi = a |Ψ1i + b |Ψ2i, then the probability of finding the system in the state |Φi will be given by 2 |hΦ|(a |Ψ1i + b |Ψ2i)| , that is, we are adding the probabilities amplitudes rather than the probabilities. This gives rise to the well-known phenomenon of quantum Key words and phrases. Quantum physics, causality, measurement, Hilbert space. 1 2 BLAKE K. WINTER interference, as well as to the well-known difficulty in interpreting the concept of measurement in quantum mechanics. 2. EPR, Bell, and causality The Einstein-Podolsky-Rosen paradox in quantum theory is a result which shows that ordinary quantum theory, assuming that our universe is a single universe, must either involve hidden variables, or else there must be a mechanism which involves superluminal causation. The original result is found in [7]. Note that the assumption that our universe is a single universe simply means that all observers involved are single observers, and they are the same observers at each point in the history considered, and it does not say anything about whether parallel universes exist or not. This assumption is denied by the many-worlds interpretation, [2], which we will discuss in Sec. 3. Theorem 1. Quantum theory in a single universe without hidden variables requires a superluminal mechanism to enforce consistent measurements, [7]. Proof. As an example, consider two spin-1/2 particles. The Hilbert space describing their spins will be the tensor product of two copies of the Hilbert space with basis |ui and |di. Thus, the Hilbert space will have a basis consisting of four vectors: |u0i |u1i, |u0i |d1i, |d0i |u1i, |d0i |d1i. Suppose these particles are produced at a certain point in spacetime and sent off in opposite directions at close to the speed of light, hitting two detectors which are separated by a spacelike interval. Suppose furthermore that their spins, prior to hitting the detectors, are described by the 1 vector ( √2 (|u0i |u1i + |d0i |d1i). Then when results are compared, either both particles are found to have been spin up, or both have been found to be spin down. But this requires either that there be local hidden variables telling the particles which state they should collapse into upon measurement, or else some superluminal mechanism to enforce their correlation. On the other hand, Bell showed that local hidden variables are incapable of reproducing the results of quantum mechanics. Theorem 2. No local hidden variable theory can reproduce the measured results of quantum mechanics. See [3] for a proof of this theorem, and [5] for an experiment demonstrating that the quantum mechanical results are experimentally verified. Bell’s theorem seems to frequently be cited as proof that hidden variables cannot be local (though non-local hidden variables, or hidden variables with superluminal causal enforcement mechanisms, can reproduce the results of quantum mechanics, e.g. [4]). However, it is often overlooked that the EPR result says that without hidden variables, there must still be some nonlocal enforcement mechanism, provided that our universe remains a single universe through history, and provided that correlations require some real causal mechanism of enforcement. 3. Relational Quantum Mechanics and Many-Worlds Here we will discuss two methods for trying to resolve the above issue. The first, relational quantum mechanics, attempts to do this in a single universe. It thus CAUSALITYANDQUANTUMTHEORY 3 ends up violating the supposition that every correlation has some real mechanism of enforcement underlying it, and appears logically unsound. The second does away with the assumption that our universe is a single universe. We begin with the notion of relational quantum mechanics, which relies on the idea that the split of the physical universe into quantum systems and classical observers is relative or relational and has been suggested by Rovelli, [10]. According to this interpretation, a state vector exists only relative to an observer. Observer A may be a part of observer B’s state vector (and vice verse). There are two problems with this approach. First, it does not define what systems will actually give rise to observers. Therefore, it depends, like the Copenhagen interpretation, on arbitrarily inserting observers into the physical universe. However, it is possible that this problem could be eventually resolved in a satisfactory manner. Second, it does not explain why two observers will agree whenever they compare experimental results. As an example, suppose there are a pair of spin-1/2 particles, a and b, which are entangled so they are both spin up or both spin down. Suppose observer A measures the spin of particle a and observer B measures the spin of particle B. Then according to observer A, measuring particle a causes the state vector to collapse to a state where particle b has exactly one spin, while observer B sees measuring particle b to cause his state vector to collapse particle a to a state with only one spin. When A and B compare their results, A sees B’s measurement as being caused by A’s state vector, while B sees A’s measurement as being caused by the collapse of B’s state vector. Rovelli argues that this is no different than the case of chronological ordering in special relativity, in which two observers may disagree about which event precedes another. Therefore, Rovelli is comfortable with the fact that A and B disagree about the cause of correlation between A’s results and B’s results. However, we maintain that the two cases are quite different. In special relativity, chronological order is no longer fixed, but there is still a well-defined causal order between events. Two observers in special relativity will agree about the causes of a chain of events, even though they might disagree on the chronological order. Ac- cording to our understanding of Rovelli’s relational quantum mechanics, however, the question ”what actually causes there to be a correlation between A’s measurements and B’s measurements” has no answer. A and B will disagree about the cause. Relational quantum mechanics is therefore incompatible with the philosoph- ical tenet that there is an objective or real cause for the correlation between the measurements of the two observers. The only method that we can see to try to salvage both the relational interpretation and the objectivity of causes is to add in a many-worlds type interpretation, [2]. In order to avoid any kind of spacelike causes, we might posit that when observer A goes to meet observer B, the act of comparing results causes observer A to branch into a universe with a compatible version of observer B. Note that we cannot say that A was already in that universe from the moment they made the first measurement, without returning to the need to enforce spacelike correlations.

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